# Iterated Eisenstein \tau-integrals and Multiple Eisenstein L-series

## TL;DR

This paper investigates the algebraic structures and relations of iterated Eisenstein tau-integrals and multiple Eisenstein L-series, establishing their independence and connections to modular values.

## Contribution

It introduces the algebraic framework for these integrals and L-series, proving their linear independence and elucidating their relations and connections to modular values.

## Key findings

- Established algebraic relations among Eisenstein tau-integrals and L-series.
- Proved linear independence of the elements within these algebraic structures.
- Connected double Eisenstein L-functions to holomorphic double modular values.

## Abstract

In this paper we study iterated Eisenstein {\tau}-integrals and multiple Eisenstein L-series, they are functions on the complex upper half plane and form two Q-algebras. They reduce to iterated Eisenstein integrals and multiple Hecke L-functions with respect to Eisenstein series respectively after analytic extension when {\tau}->0. We give the relations among them and prove the linear independence of their elements. Finally, we explain the connections among double Eisenstein L-functions and holomorphic double modular values.